3D Models by Henry Segerman

Henry Segerman is an Australian mathematician who creates 3D printed models that express mathematical formulae and concepts. This allows his students to better understand them.

“The language of mathematics is often less accessible than the language of art, but I can try to translate from one to the other, producing a picture or sculpture that expresses a mathematical idea.”

Hevea Project

A French team of mathematicians called the Hevea Project, have created digital constructions of isometric embeddings.

“Take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius,” writes Daniel Matthews about isometric embedding. “Nash and Kuiper show that by ‘ruffling’ the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius.”

My last post seemed to go down well so I thought I’d compile a few more images of proofs without words!

Determinant is the area of a parallelogram, by Solomon Golomb, Mathematics Magazine, March 1985.

A visual proof of Jensen’s inequality, found on Wikipedia. Jensen’s inequality states that

In the diagram, the dashed curve along the X axis is the hypothetical distribution of X, while the dashed curve along the Y axis is the corresponding distribution of Y values. Note that the convex mapping Y(X) increasingly “stretches” the distribution for increasing values of X.

The golden lines highlight the complexity of turbulent convective flow. They have been created using a numerical simulation of turbulent Rayleigh-Benard convection, which is when the fluid is trapped between two plates and is heated from above and cooled from below. This visualisation shows skin friction on the bottom (heated) plate in a flow of turbulently convecting liquid mercury. The bright lines are areas with large velocity changes at the wall, showing high shear stress and vigorous convective flow.

This is a tessellation of Penrose tiles. Penrose tiles can be arranged, as has been done in this image, such that the tiling never repeats, no matter how many tiles are used.The two different tile shapes are a wider rhombus and a thinner rhombus. Additionally, each tile has been filled with four pursuit curves: the dark curves from each corner to a point neat the centre of the tile.

This image was created by iterating systems of Möbius transformations. A Möbius transformation in the complex plane is a rational function of the form

This double knit scarf brings together two mathematical ideas: a recursive construction of a fractal – the dragon fractal, and the recursive construction of an integer sequence – the Fibonacci sequence. The main panels of the scarf are based on a pattern which arises from the 11th iteration of the dragon fractal, whilst the striping between the main panels illustrates the Fibonnaci sequence (colour changes after 1 row, after 1 row, after 2 rows, after 3 rows, after 5 rows etc).

This image shows 8,000 ellipses. For each k=1,2,3,…,8000 the foci of the kth ellipse are:

The eccentricity of the kth ellipse is D(k), where

This is part of the generating tile of a planar repeating pattern that have no reflection symmetries but do have many glide reflection symmetries as well as translational symmetries and two-fold centres of rotation. “The absence of reflectional symmetries often leads to very fluid and dynamic patterns.”

This image is produced by visualising the foraging behaviour of the ant P. barbatus. 1000 ants stream out of the nest along 6 main trails and break off to individually search for seeds. The image only highlights 500 of their steps. The colour graduations distinguish the “search phase from the return phase and reinforce the dynamic aspects of the process”.

“Bunny” by Craig S. Kaplan and Henry Segerman

This sculpture is self-referential; it is a sculpture of a bunny whose surface is tiled by 72 copies of the word ‘bunny’. It is an example of an autologlyph, which is “a word written or represented in a way which is described by the word itself”. This type of autologlyph combines “Escher-like tessellation with typographical ideas related to ambigrams“.

Intrinsic Transformation I by Conan Chadbourne

This work shows the structure of the icosahedral group. This is the smallest non-abelian simple group and is the set of “orientation-preserving symmetries of the regular icosahedron and dodecahedron”. The groups elements are shown as yellow disks arranged at the vertices of a truncated icosahedron, whilst the group’s generators – orders 2 and 5, which are coloured red and blue respectively – are depicted by the regions between the disks.

Visualisation of the musical piece “Five Armies” by Kevin MacLeod

Musical Flocks is a project which produces animations by simulating the behaviour of agents that react to the sound of music. Slow musics results in the slow and gentle movement of the flock, whilst fast tempo music results in fast movement and abrupt changes. Sounds which have a high volumes and a “rich” frequency spectrum affect the majority of the boids, whereas low volume and less active frequencies gives “more subtle visual variations and slower graphic evolution”.

Touch-graph by Annie Verhoeven

Verhoeven used ‘thread art’ to create the sculpture shown above. It visualises a computer program that is used to show connections between people; the coloured lines symbolise the different types of connections between people. “All these different connections are used in our life on every level, be it scientific, religious, political or otherwise, and can help us determine where we stand in the world.”

Stars of the Mind’s Sky by Paul Salomon

In this image, 300 starts are positioned along concentric circles. The number of points on the star increases as the radius from the centre increases, and the stars with the same number of points are placed evenly along their circle according to the density (“jump number”) used to generate them. Mathematically, this can be seen as representing subgroups and cosets generated by elements of a cyclic group. The different colours highlight the number of cosets, with the red stars representing the generating element. Thus, “we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.”

The Yang-Baxter scarf by Robin Endelman

The scarf above depicts the Yang-Baxter equation in statistical mechanics. A variation of this equation is used in braid theory and in the 3rd Reidemeister move in knot theory. In this scarf, the numbers 1, 2 and 3 are assigned to the colours blue, green and gold respectively. The Yang-Baxter equation states that R12 R13 R23 = R23 R13 R12, where Rij denotes the point where strand i crosses strand j;the two sides of the equation are highlighted by the two ends of the scarf, and the middle section demonstrates the equality (=).

Hyperbolic Fractal Tiling 1 by Robert Fathauer

This image shows four different views of the same 3D object – a fractal tiling where every tile has a similar dart shape. This demonstrates how a complex organic structure can be created from the repeated application of a simple set of rules to a simple starting structure.

This is the third installment of my ‘Pictures of Maths’ series. Hope you enjoy!

The structure of honeycomb displays symmetry.

The complex folding patterns that arise when layered paper is put into a test machine and squashed.

A sample image generated by Lawrence Ball’s harmonic maths.

Geometric art is often used as decoration in the ceilings of buildings.

‘Vitruvian Man’, drawn in 1487 by Leonardo Da Vinci, showed the relationship between the human body and geometry. It is a piece of art that represents how closely connected science and art are.

The 421 polytope is believed to be the most geometrically complex and aesthetically beautiful structure in mathematics. It is the algebraic form at the centre of a Universal Theory of Everything. It was originally describe in the late 19th century and models all interactions and transformations between known and theorised sub-atomic particles. The theory is an attempt to unify quantum physics and gravitation in hopes of ultimately explaining the fabric of the universe. The visualisation was hand drawn in illustrator to an accuracy of 1/10,000 of a millimetre.

The Klein Bottle: an object with no boundaries, no inside or outside. It is a one sided, non-orientable surface. That’s topology for you!

Fractal patterns are visible throughout nature, for example in ferns as displayed above.

A 3D print of the tesseract, which is the four dimensional analogue of the cube.

Borromean rings is an arrangement of three topological circular rings which are linked in such a fashion that removing any ring results in two un-linked rings. The name comes from their use in the coat of arms of the aristocratic Borromeo family in Northern Italy.

Penrose tiling, named after mathematician Roger Penrose, is made from a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders).

Projective curve in ℙ^2

The Brillouin zones of a square crystal lattice in two dimensions, which underlie the analysis of waves propagating through the crystal.

This image represents the loxodrome of a sphere. A loxodrome is the path taken when a compass is kept pointing in a constant direction. If the surface is a sphere, the loxodrome is a spherical spiral.